Average Error: 9.6 → 0.3
Time: 12.3s
Precision: 64
\[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
↓
\[\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]
\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}
↓
\frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}
double f(double x) {
        double r1691119 = 1.0;
        double r1691120 = x;
        double r1691121 = r1691120 + r1691119;
        double r1691122 = r1691119 / r1691121;
        double r1691123 = 2.0;
        double r1691124 = r1691123 / r1691120;
        double r1691125 = r1691122 - r1691124;
        double r1691126 = r1691120 - r1691119;
        double r1691127 = r1691119 / r1691126;
        double r1691128 = r1691125 + r1691127;
        return r1691128;
}
↓
double f(double x) {
        double r1691129 = 2.0;
        double r1691130 = x;
        double r1691131 = 1.0;
        double r1691132 = r1691130 + r1691131;
        double r1691133 = r1691132 * r1691130;
        double r1691134 = r1691130 - r1691131;
        double r1691135 = r1691133 * r1691134;
        double r1691136 = r1691129 / r1691135;
        return r1691136;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original9.6
Target0.3
Herbie0.3
\[\frac{2}{x \cdot \left(x \cdot x - 1\right)}\]

Derivation

  1. Initial program 9.6

    \[\left(\frac{1}{x + 1} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  2. Using strategy rm

  3. Applied add-cube-cbrt24.3

    \[\leadsto \left(\frac{1}{\color{blue}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

  4. Applied *-un-lft-identity24.3

    \[\leadsto \left(\frac{\color{blue}{1 \cdot 1}}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

  5. Applied times-frac25.2

    \[\leadsto \left(\color{blue}{\frac{1}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}} \cdot \frac{1}{\sqrt[3]{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]
  6. Using strategy rm

  7. Applied frac-times24.3

    \[\leadsto \left(\color{blue}{\frac{1 \cdot 1}{\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}}} - \frac{2}{x}\right) + \frac{1}{x - 1}\]

  8. Applied frac-sub28.7

    \[\leadsto \color{blue}{\frac{\left(1 \cdot 1\right) \cdot x - \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) \cdot 2}{\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) \cdot x}} + \frac{1}{x - 1}\]

  9. Applied frac-add25.5

    \[\leadsto \color{blue}{\frac{\left(\left(1 \cdot 1\right) \cdot x - \left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) \cdot 2\right) \cdot \left(x - 1\right) + \left(\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) \cdot x\right) \cdot 1}{\left(\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

  10. Simplified25.2

    \[\leadsto \frac{\color{blue}{\left(x + 1\right) \cdot x + \left(x - 1\right) \cdot \left(x - 2 \cdot \left(x + 1\right)\right)}}{\left(\left(\left(\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}\right) \cdot \sqrt[3]{x + 1}\right) \cdot x\right) \cdot \left(x - 1\right)}\]

  11. Simplified25.1

    \[\leadsto \frac{\left(x + 1\right) \cdot x + \left(x - 1\right) \cdot \left(x - 2 \cdot \left(x + 1\right)\right)}{\color{blue}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}}\]

  12. Taylor expanded around 0 0.3

    \[\leadsto \frac{\color{blue}{2}}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

  13. Final simplification0.3

    \[\leadsto \frac{2}{\left(\left(x + 1\right) \cdot x\right) \cdot \left(x - 1\right)}\]

Reproduce

herbie shell --seed 2019154 
(FPCore (x)
  :name "3frac (problem 3.3.3)"

  :herbie-target
  (/ 2 (* x (- (* x x) 1)))

  (+ (- (/ 1 (+ x 1)) (/ 2 x)) (/ 1 (- x 1))))